Qualitative theory of three-dimensional rotator model

The 3-dimensional planar rotator (PR) model with short-range interaction is studied in the “harmonic” approximation where spin wave and vortex loop (VL) system decouple. The phase transition (PT) occuring in the PR-model is studied via the correlation and stability properties of the VL-system for which a tentative phase diagram is set up. Under the assumption that the PT of the PR-model is continuous an ansatz for the “dielectric” function ?_q of the VL-system is made which leads at criticality simultaneously to scaling of the VL-system and of the PR-model. Physical arguments are given to justify the dimensional properties of ?_q which led to this result. The critical exponents for the VL-system in usual notation are η_L=1, and γ_L=v_L=0. The latter represent logarithmic singularities, i.e., ?₀～–ln|τ|^z_γ, and κ～–1/ln|τ|^z_v. The critical expon ents of the PR-model can be expressed in terms of those of the VL-system. The low and high-temperature correlation functions are given. It is shown that only in the presence of an anisotropy field, e.g., a magnetic field, the low temperature correlation function decays exponentially. The relation between the PR-model, and the melting of crystals is briefly discussed.